Make animated particle object start animation from the beginning when emitted?
I'm sorry to say that this feature, which was started on in 2011, was "nearly complete" in 2013 yet still seems like a distant dream. Some of the changes necessary for these updates were dependent on the dependency graph update, though. The depsgraph branch has had a lot of progress lately. Perhaps reading about dependencies will explain why things like this don't presently work out of the box; e.g. why you still can't "animate everything" as the 2.5 project advertised.

The latest update I can find in the same vein is that Lukas wrote a patch that allows you to use a particle's age to update a shader. 

The best thing you can do to participate right now with in-progress projects like this is to get in contact with the specific developers that are working on the relevant infrastructures, and show them the problem you're trying to solve. Let them know you're interested, try their test branches, test out experimental versions, etc, and if you're feeling bold,  look at their code.